It would have to be a very distracted person, as you say, to not realise granny hadn't paid her bills. Also I think it would creep a few people out to be asked to say "I love you mom" to a stranger knowing that she thought you looked like her dead daughter. That's just a little sick IMO. Having said that, I do think some people might feel sorry for her and have trouble saying no.
I would have been very uncomfortable with the whole thing. I certainly would have been compassionate in the beginning, but as things went on, it was a bit strange. I would have avoided the woman at some point.
I would have ignored her. Only because she kept on following the gal around and that is odd. I would not call her mom. I sound terrible don't I?
One time I know I did help a lady when I lived in Wyoming. She was trying to get her prescription drugs and she was elderly and only received a pension check in the mail. The prescription people know her well, but they could not help her. She used the last of her money which was $20. I saw her wallet in her purse and other things in there, and I know she needed help for real. The pharmacist said that she either gets the prescription and does not have food the rest of the month until her next pension check arrives. I paid for her prescription.
Another time, another elderly lady just needed someone to talk to. I talked with her in the store parking lot for over 2 hours. She thanked me over and over again.
The Granny Scam...I have grey hair now, maybe this will work for me! Just kidding!